Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces

Volume 11, Issue 8, pp 984--993 http://dx.doi.org/10.22436/jnsa.011.08.05
Publication Date: June 09, 2018 Submission Date: April 05, 2018 Revision Date: April 22, 2018 Accteptance Date: April 26, 2018

Authors

Ahmet Eroglu - Department of Mathematics, Nigde Omer Halisdemir University, Nigde, Turkey. Tahir Gadjiev - Institute of Mathematics and Mechanics, NAS of Azerbaijan, AZ1141 Baku, Azerbaijan. Faig Namazov - Baku State University, AZ1141 Baku, Azerbaijan.


Abstract

Let \(L=-\Delta_{\mathbb{H}_n}+V\) be a Schrödinger operator on the Heisenberg groups \(\mathbb{H}_n\), where the non-negative potential \(V\) belongs to the reverse Hölder class \(RH_{Q/2}\) and \(Q\) is the homogeneous dimension of \(\mathbb{H}_n\). Let \(b\) belong to a new \(BMO_{\theta}(\mathbb{H}_n,\rho)\) space, and let \({\cal I}_{\beta}^{L}\) be the fractional integral operator associated with \(L\). In this paper, we study the boundedness of the operator \({\cal I}_{\beta}^{L}\) and its commutators \([b,{\cal I}_{\beta}^{L}]\) with \(b \in BMO_{\theta}(\mathbb{H}_n,\rho)\) on central generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) and generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) associated with Schrödinger operator. We find the sufficient conditions on the pair \((\varphi_1,\varphi_2)\) which ensures the boundedness of the operator \({\cal I}_{\beta}^{L}\) from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(M_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\), \(1/p-1/q=\beta/Q\). When \(b\) belongs to \(BMO_{\theta}(\mathbb{H}_n,\rho)\) and \((\varphi_1,\varphi_2)\) satisfies some conditions, we also show that the commutator operator \([b,{\cal I}_{\beta}^{L}]\) is bounded from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}\) to \(M_{q,\varphi_2}^{\alpha,V}\), \(1/p-1/q=\beta/Q\).


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ISRP Style

Ahmet Eroglu, Tahir Gadjiev, Faig Namazov, Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 984--993

AMA Style

Eroglu Ahmet, Gadjiev Tahir, Namazov Faig, Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces. J. Nonlinear Sci. Appl. (2018); 11(8):984--993

Chicago/Turabian Style

Eroglu, Ahmet, Gadjiev, Tahir, Namazov, Faig. "Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 984--993


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